0$ for $x<0$, and $g'(x)<0$ for $x>0$. Since $g(x)-x$ is positive for $x=0$ and negative for $x=1$, there is a unique fixed point $p>0$, and $g(-p)=p$. Clearly ${\cal R}g(\pm 1) = - g^{2}(\pm p)/p=-1$. For $x \in (-p,p)$ there is $g(x)>p$, so $g'(g(x))<0$. The equation $(g^{2})'(x)= g'(g(x)) g'(x)=0$ has in $-p,p!$ unique solution $0$, so the equation $({\cal R}g)'(x)=0$ has in $-1,1!$ the unique solution $0$. Since $g$ is decreasing for $x>0$, and $0

p$, and $g^{2}(0)

-1$. $\Box$
\begin{thm} \label{renorm}
Let $g$ be a symmetric unimodal system with $g(0)>0$. If ${\cal
R}g(0)\leq 1$, and if ${\cal R}g$ is a factor of a finite
automaton, then $g$ is also a factor of a finite automaton.
\end{thm}
\begin{picture}(200,60)
\put(65,45){\arleft{f}{X}}
\put(65,15){\ardown{\varphi}{\varphi}}
\put(65,5) {\arleft{g^{2}}{I_{0}}}
\put(205,45){\arleft{h}{X''}}
\put(205,15){\ardown{\psi}{\psi}}
\put(205,5) {\arleft{g}{I}}
\end{picture}
\noindent
Proof: Let $p_{0}$ be the positive fixed point of $g$. There is
an increasing sequence $p_{k} \in (0,1)$ such that $g(-p_{k})$ =
$g(p_{k})$ = $-p_{k-1}$ for $k>0$. Denote $I_{-\infty}=\{-1\}$,
$I_{k}=-p_{-k},-p_{-k+1}!$ for $k<0$, $I_{0}=-p_{0},p_{0}!$,
$I_{k}=p_{k-1},p_{k}!$ for $k>0$, and $I_{\infty}=\{1\}$. Then
$g(I_{-k}) = g(I_{k}) = I_{-k+1}$ for $k>0$. Since ${\cal R}g(I)
\subseteq I$, we have $g^{2}(I_{0}) \subseteq I_{0}$, and
$g(I_{0}) \subseteq I_{1}$. By the assumption there exists
a finite automaton $(X,f)$ and a factorization $\varphi:(X,f)
\rightarrow (I_{0},g^{2})$. We construct a finite automaton
$(X',h)$. Suppose $A=\{0,...,m-1\}$ and define $A'=\{0,...,m\}$,
$Q'=\{(q,i): q \in Q, i\in \{-1,0,1,\sharp\}\}$, $B'=B$. For $x
\in Q'$ denote $\lfloor x \rfloor$ its first component, and
$\lceil x \rceil$ the second, so that $x=(\lfloor x
\rfloor,\lceil x \rceil)$. Define
\ \begin{array}{ll}
X_{-\infty} & = \{u \in X':\; \lceil u_{0} \rceil=-1,
(\forall j >0)(u_{j}=m) \}, \\
X_{k} & = \{u \in X':\; \lceil u_{0} \rceil = \mbox{sgn}(k),
\min\{j > 0:u_{j} \neq m\} = :k: \},\;\; :k: > 0 \\
X_{\infty} & = \{u \in X':\; \lceil u_{0} \rceil=1, (\forall j
>0)(u_{j}=m) \} \\
X_{i} & = \{u \in X':\; \lceil u_{0} \rceil = i \} \;\;
\mbox{ for }\;\; i \in \{0,\sharp\} \end{array} \!
Define $h_{B} : Q' \rightarrow B'$, $h_{A}: Q' \times A'
\rightarrow Q'$ by $h_{B}(x) = f_{B}(\lfloor x \rfloor)$, \
\begin{array}{lll}
h_{A}(x,m) & = (\lfloor x \rfloor, -1) & \mbox{ if } \;
\lceil x \rceil \in \{-1,1\} \\
h_{A}(x,j) & = (\lfloor x \rfloor, 0) & \mbox{ if } \;
\lceil x \rceil \in \{-1,1,\sharp\},\; j 0$, then there exists ${\cal R}(g)$ and $I$ is
its invariant set (otherwise $g$ would have infinite number of
periodic points). If ${\cal R}g(0)>0$, we construct further
renormalizations. Since $g$ has only a finite number of periodic
points, there exists $k \geq 0$, with ${\cal R}^{k}g(0) \leq 0$.
By Proposition \ref{negative}, ${\cal R}^{k}g$ is a factor of a
finite automaton, and by Theorem \ref{renorm}, so is $g$. $\Box$
\begin{pro}
Let $(I,g)$ be a symmetric unimodal system with $g(0)=1$. Then
$g$ is a factor of a finite automaton.
\end{pro}
Proof: Define a closed cover $\alpha= \{-1,0!,0,1!\}$, and a
mapping $\varphi: 2^{N} \rightarrow I$ by
$\varphi(u) = x \in I$ iff $(\forall i \geq 0)(g^{i}(x) \in
\alpha_{i} \}$. It follows from the results in Collet and Eckmann
\cite{kn:Collet}, that $\varphi$ is a surjective mapping, and
$\varphi: (2^{N},\sigma) \rightarrow (I,g)$ is a factorization.
$\Box$
\begin{cor}
Let $(I,g)$ be a symmetric unimodal system at a band-merging
bifurcation, i.e. ${\cal R}^{k}g(0)=1$ for some $k$. Then it is a
factor of a finite automaton.
\end{cor}
\begin{exm}
A symmetric unimodal system at the common limit of period
doubling and band merging bifurcations has not chaotic limits.
\end{exm}
Proof: There exists an infinite closed invariant set $Y \subseteq
I$, which is topologically transitive but has no periodic point.
The set of periodic points $P \subseteq I$ is infinite too, and
its closure is $P \cup Y$. Moreover, for any finite $A \subseteq
P$ there exists $U \subseteq I$ with $Y \subseteq U$ and
$\omega(U) \cap A = 0$ (see Collet and Eckmann
\cite{kn:Collet} and Falconer \cite{kn:Fal} for more details).
Let $y \in Y$ and $y \in A \subseteq I$. If $\omega(A) \subseteq
Y$, then it has no periodic point. If $\omega(A) \cap P \neq 0$,
then $\omega(A)$ is not topologically transitive. In either case
$\omega(A)$ is not chaotic. $\Box$
\pagebreak
\begin{thebibliography}{99}
\bibitem{kn:Balcar} B.Balcar, P.Simon: Appendix on general
topology. Handbook of Boolean Algebras (J.D.Monk and R.Bonnet,
eds.), Elsevier Science Publishers B.V., (1989) 1241.
\bibitem{kn:Brooks} J.Brooks, G.Cairns, G.Davis, P.Stacey: On
Devaney's definition of chaos. The American Mathematical Monthly,
99,4 (1992) 332.
\bibitem{kn:Collet} P.Collet, J.P.Eckmann.: Iterated Maps on
the Interval as Dynamical Systems. Birkhauser, Basel (1980).
\bibitem{kn:Crutch} J.P.Crutchfield, K.Young: Computation at the
onset of chaos. in: Complexity, Entropy and the Physics of
Information, SFI Studies in the Sciences of Complexity, vol VIII
(W.H.Zurek, ed.), Addison Wesley (1990) 223.
\bibitem{kn:Devaney} R.L.Devaney: An Introduction to Chaotic
Dynamical Systems. Addison-Wesley, Redwood City (1989).
\bibitem{kn:Fal} K.J.Falconer: The Geometry of Fractal Sets.
Cambridge University Press, Cambridge (1985).
\bibitem{kn:Hop} J.E.Hopcroft, J.D.Ullmann: Introduction to
Automata Theory, Languages, and Computation. Addison-Wesley,
Menlo Park (1990).
\bibitem{kn:Langton} Ch.G.Langton: Life at the edge of chaos.
in: Artificial Life II. SFI Studies in the Sciences of
Complexity, vol. X (Ch.G.Langton, Ch.Taylor, J.D.Farmer,
S.Rasmunsen, eds.) Addison-Wesley, Redwood City (1992).
\bibitem{kn:Shub} M.Shub: Global Stability of Dynamical
Systems. Springer-Verlag, Berlin (1987).
\bibitem{kn:Troll} J.Troll: Formal language characterization
of transitions to chaos of truncated horseshoes. Technische
Universit\"{a}t Berlin, SFB 288, No 13, 1992.
\end{thebibliography}
\end{document}